The Giant Badger has a burrow speed of 10'. Since your movement speed while dragging an enemy is halved, this means you can put your enemy 5' underground without using an action.
This addresses what happens in 3.5e, but I don't have a 5e PHB in front of me at the moment. I can imagine two situations – the burrowing rules have changes and a badger leaves a tunnel, or they haven't.
If there is a tunnel, the target can crawl through it and return to the surface (although probably prone – a definite advantage over other grapplers, who spend either a small a attack or bonus action w/ shield master to prone)
If not, the target will be left underground, and begin suffocating unless it can dig through the dirt fast enough. (Are there rules for this in 5e?)
What happens in 5th edition.